orthogonal group

Definitions

1. 1

   For given n and field F (especially where F is the real numbers), the group of n × n…

   For given n and field F (especially where F is the real numbers), the group of n × n orthogonal matrices with elements in F, where the group operation is matrix multiplication.

   • The symbol O^ε(n,q) for orthogonal groups has been well established in finite group theory as and, throughout the mathematics community, O(n, K) stands for an orthogonal group when K is the real or complex field.
   • Theorem 7.4. Let n ≥ 1. For odd primes p#92;not#92;equiv#92;pm 1#92;pmod#123;24#125; the odd-dimensional orthogonal groups O#95;#123;2n#43;1#125;(p) possess GA-realizations over #92;mathbb#123;Q#125;.
   • The normalizer of the full orthogonal group within the general linear group can be shown to consist of all (commutative) products of spherical dilatations and orthogonal transformations.

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